No, flight crews do not continually monitor transponder operation. It is not a navigation instrument.
If it fails or is turned off, usually ATC will be the first to notice and ask the aircraft to confirm it is on.
As far as I know it was clear to the ATCs in Brasilia that the Legacy transponder was off and that the data they were reading were not trustworthy. This lasted for almos one hour and...no attemps were made to cintact the Legacy and informing them that their transponder was off. I agree with you.
Insofar as the Legacy pilots, in hindsight (pay attention...hindsight...) they could have done two things that could prevent the disaster: question the change in the flight plan to 370 to Manaus and see in their flight data screen the small yellow letters saying that the transponder was off. I said..in hindsight...knowing what happened...
We canīt make a statement that any ATC in every country will need the ATCs of planes to prevent air collisions... The system should be able to avoid collisions without ATCs...